Permutation Matrix Generation
The standard form of a Cayley table has the order of the elements in the rows the same as the order in the columns. Another form is to arrange the elements of the columns so that the nth column corresponds to the inverse of the element in the nth row. In our example of D3, we need only switch the last two columns, since f and d are the only elements that are not their own inverses, but instead inverses of each other.
| e | a | b | c | f=d-1 | d=f-1 | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | f | d |
| a | a | e | d | f | c | b |
| b | b | f | e | d | a | c |
| c | c | d | f | e | b | a |
| d | d | c | a | b | e | f |
| f | f | b | c | a | d | e |
This particular example lets us create six permutation matrices (all elements 1 or 0, exactly one 1 in each row and column). The 6x6 matrix representing an element will have a 1 in every position that has the letter of the element in the Cayley table and a zero in every other position, the Kronecker delta function for that symbol. (Note that e is in every position down the main diagonal, which gives us the identity matrix for 6x6 matrices in this case, as we would expect.) Here is the matrix that represents our element a, for example.
| e | a | b | c | f | d | |
|---|---|---|---|---|---|---|
| e | 0 | 1 | 0 | 0 | 0 | 0 |
| a | 1 | 0 | 0 | 0 | 0 | 0 |
| b | 0 | 0 | 0 | 0 | 1 | 0 |
| c | 0 | 0 | 0 | 0 | 0 | 1 |
| d | 0 | 0 | 1 | 0 | 0 | 0 |
| f | 0 | 0 | 0 | 1 | 0 | 0 |
This shows us directly that any group of order n is a subgroup of the permutation group Sn, order n!.
Read more about this topic: Cayley Table
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